YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRS Reverse [EQUIVALENT, 0 ms]
(2) QTRS
(3) FlatCCProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RootLabelingProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 55 ms]
(8) QTRS
(9) DependencyPairsProof [EQUIVALENT, 8 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) QDP
(13) QDPOrderProof [EQUIVALENT, 40 ms]
(14) QDP
(15) QDPOrderProof [EQUIVALENT, 193 ms]
(16) QDP
(17) DependencyGraphProof [EQUIVALENT, 0 ms]
(18) QDP
(19) QDPOrderProof [EQUIVALENT, 23 ms]
(20) QDP
(21) QDPOrderProof [EQUIVALENT, 618 ms]
(22) QDP
(23) DependencyGraphProof [EQUIVALENT, 0 ms]
(24) TRUE


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(x1) -> b(c(x1))
   b(a(b(x1))) -> x1
   c(c(x1)) -> a(a(a(b(x1))))

Q is empty.

----------------------------------------

(1) QTRS Reverse (EQUIVALENT)
We applied the QTRS Reverse Processor [REVERSE].
----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(x1) -> c(b(x1))
   b(a(b(x1))) -> x1
   c(c(x1)) -> b(a(a(a(x1))))

Q is empty.

----------------------------------------

(3) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]
As Q is empty the flat context closure was sound AND complete.

----------------------------------------

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a(a(x1)) -> a(c(b(x1)))
   c(a(x1)) -> c(c(b(x1)))
   b(a(x1)) -> b(c(b(x1)))
   a(b(a(b(x1)))) -> a(x1)
   c(b(a(b(x1)))) -> c(x1)
   b(b(a(b(x1)))) -> b(x1)
   a(c(c(x1))) -> a(b(a(a(a(x1)))))
   c(c(c(x1))) -> c(b(a(a(a(x1)))))
   b(c(c(x1))) -> b(b(a(a(a(x1)))))

Q is empty.

----------------------------------------

(5) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled

As Q is empty the root labeling was sound AND complete.

----------------------------------------

(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(x1)
   a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(x1)
   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(x1)
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(x1)
   b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(x1)
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(x1)
   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))

Q is empty.

----------------------------------------

(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(a_{a_1}(x_1)) = 1 + x_1
   POL(a_{b_1}(x_1)) = 1 + x_1
   POL(a_{c_1}(x_1)) = 2 + x_1
   POL(b_{a_1}(x_1)) = x_1
   POL(b_{b_1}(x_1)) = x_1
   POL(b_{c_1}(x_1)) = 1 + x_1
   POL(c_{a_1}(x_1)) = 1 + x_1
   POL(c_{b_1}(x_1)) = x_1
   POL(c_{c_1}(x_1)) = 2 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(x1)
   a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(x1)
   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(x1)
   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(x1)
   b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(x1)
   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(x1)
   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))
   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))))




----------------------------------------

(8)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))

Q is empty.

----------------------------------------

(9) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
   A_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
   A_{A_1}(a_{b_1}(x1)) -> A_{C_1}(c_{b_1}(b_{b_1}(x1)))
   A_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   C_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
   C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
   B_{A_1}(a_{c_1}(x1)) -> C_{B_1}(b_{c_1}(x1))
   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
   B_{A_1}(a_{b_1}(x1)) -> B_{C_1}(c_{b_1}(b_{b_1}(x1)))
   B_{A_1}(a_{b_1}(x1)) -> C_{B_1}(b_{b_1}(x1))
   C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(x1)
   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1))
   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{b_1}(x1))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes.
----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(x1)
   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   A_{A_1}(a_{c_1}(x1)) -> A_{C_1}(c_{b_1}(b_{c_1}(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(A_{A_1}(x_1)) = 1
   POL(A_{C_1}(x_1)) = x_1
   POL(B_{A_1}(x_1)) = 1
   POL(B_{C_1}(x_1)) = 1
   POL(C_{A_1}(x_1)) = 1
   POL(C_{B_1}(x_1)) = 1
   POL(a_{a_1}(x_1)) = 0
   POL(a_{b_1}(x_1)) = 0
   POL(a_{c_1}(x_1)) = 0
   POL(b_{a_1}(x_1)) = 1
   POL(b_{b_1}(x_1)) = 0
   POL(b_{c_1}(x_1)) = 0
   POL(c_{a_1}(x_1)) = 1
   POL(c_{b_1}(x_1)) = x_1
   POL(c_{c_1}(x_1)) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))


----------------------------------------

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(x1)
   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(15) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   C_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> C_{A_1}(x1)
   C_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   C_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   A_{A_1}(a_{a_1}(x1)) -> A_{C_1}(c_{b_1}(b_{a_1}(x1)))
   A_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   A_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
   A_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
   C_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( A_{A_1}_1(x_1) ) = x_1 + 1
POL( B_{A_1}_1(x_1) ) = x_1
POL( C_{B_1}_1(x_1) ) = x_1
POL( b_{a_1}_1(x_1) ) = x_1 + 1
POL( b_{b_1}_1(x_1) ) = max{0, x_1 - 2}
POL( A_{C_1}_1(x_1) ) = x_1
POL( c_{b_1}_1(x_1) ) = x_1
POL( B_{C_1}_1(x_1) ) = x_1
POL( a_{a_1}_1(x_1) ) = x_1 + 1
POL( a_{b_1}_1(x_1) ) = x_1
POL( a_{c_1}_1(x_1) ) = x_1 + 1
POL( b_{c_1}_1(x_1) ) = x_1 + 1
POL( c_{a_1}_1(x_1) ) = x_1 + 2
POL( c_{c_1}_1(x_1) ) = x_1 + 2
POL( C_{A_1}_1(x_1) ) = x_1 + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))


----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   B_{A_1}(a_{a_1}(x1)) -> C_{B_1}(b_{a_1}(x1))
   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
   A_{C_1}(c_{c_1}(c_{b_1}(x1))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))
   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(19) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(c_{b_1}(b_{c_1}(x1)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial interpretation [POLO]:

   POL(B_{A_1}(x_1)) = 1
   POL(B_{C_1}(x_1)) = x_1
   POL(a_{a_1}(x_1)) = 0
   POL(a_{b_1}(x_1)) = 0
   POL(a_{c_1}(x_1)) = 0
   POL(b_{a_1}(x_1)) = 1
   POL(b_{b_1}(x_1)) = 0
   POL(b_{c_1}(x_1)) = 0
   POL(c_{a_1}(x_1)) = 1
   POL(c_{b_1}(x_1)) = x_1
   POL(c_{c_1}(x_1)) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))


----------------------------------------

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(21) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   B_{C_1}(c_{c_1}(c_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(B_{C_1}(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(c_{c_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, -I], [-I, 0A, -I], [-I, 1A, -I]] 	* 	x_1
>>>

   <<<
 POL(c_{b_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [-I, 1A, -I]] 	* 	x_1
>>>

   <<<
 POL(B_{A_1}(x_1)) =  	[[-I]] 	 +  	[[0A, 1A, -I]] 	* 	x_1
>>>

   <<<
 POL(a_{a_1}(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, -I, 0A], [-I, 1A, -I]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [-I, -I, -I], [0A, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, -I, -I], [-I, -I, 0A], [0A, 0A, 0A]] 	* 	x_1
>>>

   <<<
 POL(a_{c_1}(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, -I], [-I, -I, -I], [-I, -I, 0A]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[-I], [-I], [-I]] 	 +  	[[0A, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
>>>

   <<<
 POL(b_{c_1}(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, -I], [-I, -I, -I], [0A, 0A, -I]] 	* 	x_1
>>>

   <<<
 POL(c_{a_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[0A, -I, -I], [0A, -I, 0A], [1A, -I, -I]] 	* 	x_1
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))


----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   B_{A_1}(a_{a_1}(x1)) -> B_{C_1}(c_{b_1}(b_{a_1}(x1)))

The TRS R consists of the following rules:

   a_{a_1}(a_{a_1}(x1)) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
   a_{a_1}(a_{b_1}(x1)) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{a_1}(a_{a_1}(x1)) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
   c_{a_1}(a_{b_1}(x1)) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
   b_{a_1}(a_{a_1}(x1)) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
   b_{a_1}(a_{b_1}(x1)) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
   c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(x1)
   c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(x1)
   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(23) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(24)
TRUE